Palgrave Handbook of Econometrics: Applied Econometrics

William Greene 493

whereGis an unknown continuous distribution function whose range is [0,1].
The functionGis not specifieda priori; it is estimated (pointwise) along with the
parameters. (SinceGas well asθis to be estimated, a constant term is not identified;
essentially,Gprovides the location for the index that would otherwise be pro-
vided by a constant.) The criterion function for estimation, in whichnsubscripts
denote estimators based on the sample ofnobservations of their unsubscripted
counterparts, is:

lnLn=

1

n

∑n

i= 1

{dilnGn(w′iθn)+( 1 −di)ln[ 1 −Gn(w′iθn)]}.

The estimator of the probability function,Gn, is computed at each iteration using
a nonparametric kernel estimator of the density ofw′iθn. For the Klein and Spady
estimator, the nonparametric regression estimator is:

Gn(zi)=

dgn(zi|di= 1 )

dgn(zi|di= 1 )+( 1 −d)gn(zi|di= 0 )

,

wheregn(zi|di)is thekernel estimate of the densityofzi=w′iθn. This result is:

gn(zi|di= 1 )=

1

ndhn

∑n

j= 1

djK

(

zi−w′jθn

hn

)

;

gn(zi|di=0) is obtained by replacingdwith 1−din the leading scalar anddjwith
1 −djin the summation. The scalarhnis the bandwidth. There is no firm theory for
choosing the kernel function or the bandwidth. Both Horowitz and Gerfin used the
standard normal density. Two different methods for choosing the bandwidth are
suggested by them. Klein and Spady provide theoretical background for computing
asymptotic standard errors.
Manski’s (1975, 1985, 1986, 1987) maximum score estimator is even less param-
eterized than Klein and Spady’s model. The estimator is based on the fitting
rule:

MaximizeθSNα(θ)=

1

n

∑n

i= 1

[qi−( 1 − 2 α)]sign(w′iθ).^11

The parameterαis a preset quantile, andqi= 2 di−1 as before. Ifαis set to 0.5,
then the maximum score estimator chooses theθto maximize the number of times
that the prediction has the same sign asz. This result matches our prediction rule
in equation (11.3) withP∗=0.5. So forα=0.5, the maximum score attempts
to maximize the number of correct predictions. Since the sign ofw′θis the same
for all positive multiples ofθ, the estimator is computed subject to the constraint
thatθ′θ=1. Variants of semiparametric estimators are discussed in Li and Racine
(2007), including a modification by Horowitz (1992) and an estimator suggested
by Lewbel (2000).
The semiparametric estimators ofθare robust to variation in the distribution
of the random elements in the model, and even to heteroskedasticity. Robustness